Let $T:\mathbb{R}^2\to \mathbb{R}^2$ be a linear transformation such that $$(a,b)\longmapsto (a+b, a-b)$$ Find all the eigenvalues and, for each eigenvalue, find the corresponding eigenspace.
My attempt:
I don't know if there is a normal procedure to find the matrix of a linear transformation, but I just "back filled" the entry values to make it work. So I have $$ \begin{pmatrix} 1 & 1 \\ 1 & -1 \\ \end{pmatrix} \begin{pmatrix} a \\ b \\ \end{pmatrix}= \begin{pmatrix} a+b \\ a-b \\ \end{pmatrix} $$ So, denoting the matrix as $A$, I used the characteristic polynomial $$ det(A-\lambda I)= \begin{pmatrix} 1-\lambda & 1 \\ 1 & -1-\lambda \\ \end{pmatrix}=0 $$ $\implies -(1-\lambda)^2-1=0\implies \lambda= 1+i$ or $1-i$.
pluging the former value into the matrix I solve for $$ \begin{pmatrix} -i & 1\\ 1 & -2-i \\ \end{pmatrix} \begin{pmatrix} a \\ b \\ \end{pmatrix}= \begin{pmatrix} 0 \\ 0 \\ \end{pmatrix} $$ Which generates the system of equations $$-ia+b=0, \quad a-2b-ib=0 $$ But solving the system gives me $a=b=0$.
There is a previous problem where I got the same thing, so I'm wondering if I am doing something wrong. Any help is much appreciated.
The eigen values are $\pm \sqrt 2$.